Problem: All of the 3rd grade teachers and students from Springer went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$3.50$ each for students, and the group paid $$69.00$ in total. The next month, the same group visited a science museum where the tickets cost $$25.50$ each for teachers and $$11.00$ each for students, and the group paid $$212.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+3.5y = 69}$ ${25.5x+11y = 212}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-25.5x-10.5y = -207}$ ${25.5x+11y = 212}$ Add the top and bottom equations together. $ 0.5y = 5 $ $ y = \dfrac{5}{0.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {8.5x+3.5y = 69}$ to find $x$ ${8.5x + 3.5}{(10)}{= 69}$ $8.5x+35 = 69$ $8.5x = 34$ $x = \dfrac{34}{8.5}$ ${x = 4}$ You can also plug ${y = 10}$ into $ {25.5x+11y = 212}$ and get the same answer for $x$ ${25.5x + 11}{(10)}{= 212}$ ${x = 4}$ There were $4$ teachers and $10$ students on the field trips.